Evolution equations using derivatives of fractional order like Caputo's derivative or Riemann-Liouville's derivative have been intensively analyzed in numerous works. But the classical bounded perturbation theorem has been proven not to be in general true for these models, especially for solution operators of evolution equations with fractional order derivative $ \alpha $ less than $ 1 $ ($ 0<\alpha<1 $), as shown by the example in the next section. This paper proposes an alternative way of dealing with this issue. We make use of the conventional time derivative with a new parameter to show the perturbations by bounded linear operators for linear evolution equations when the derivative order is less than one. The new parameter which happens to be fractional, characterizes the so-called $ \beta $-derivative. Its fractional order parameter allows the use of concepts like revamped time to provide a relation between both strongly continuous two-parameter solution operators involved in the perturbation process. To validate the theory, we use an application to population dynamics and perform some numerical simulations that reveal some consistency with the expected results.

中文翻译：

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具有参数的进化方程的有界扰动及其在种群动力学中的应用

许多著作都对使用分数阶导数（如 Caputo 导数或 Riemann-Liouville 导数）的演化方程进行了深入分析。但是经典的有界微扰定理已被证明对这些模型通常不正确，特别是对于分数阶导数 $\alpha $ 小于 $1 $ ($ 0<\alpha<1 $) 的演化方程的解算符，如下一节中的示例所示。本文提出了一种替代方法来处理这个问题。我们利用带有新参数的传统时间导数来显示当导数阶数小于 1 时线性演化方程的有界线性算子的扰动。恰好是小数的新参数表征了所谓的 $\beta $-导数。它的分数阶参数允许使用改进时间等概念来提供扰动过程中涉及的两个强连续双参数求解算子之间的关系。为了验证该理论，我们将应用程序用于种群动态并执行一些数值模拟，以显示与预期结果的某些一致性。